Parry Circle

Author

Eric W. Weisstein
October 21, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/ParryCircle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ParryCircle.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Definition

t=Triangle[{{0,0},{1,0},{.3,.2}}];
Show[Graphics[{
t,
{Red,ParryCircle[t]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
{
{IsodynamicPoint[1][t],"S",Red},
{IsodynamicPoint[2][t],"S'",Red},
{Centroid[t],"G",Blue}
}
],
{PointSize[.025],LabelOffset->{1,-1}.04}
]
}],AspectRatio->Automatic,PlotRange->All]

Common Chord

CollinearQ[{ParryPoint,X[110],SymmedianPoint}]//Timing

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Red,ParryCircle[t]},
{Blue,Circumcircle[t]},
{Purple,Line[Coordinates/@{ParryPoint[t],X[110][t]}]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
{
{IsodynamicPoint[1][t],"S",Red},
{IsodynamicPoint[2][t],"S'",Red},
{Centroid[t],"G",Red},
{ParryPoint[t],"P",Purple},
{X[110][t],"(Subscript[X, 110])",Purple},
{SymmedianPoint[t],"K",Purple},
{ParryCenter[t],"(Subscript[O, P])",Red}
}
],
{PointSize[.04],LabelOffset->-.08}
]
}],AspectRatio->Automatic,PlotRange->All]

Equation

CircleFunction[ParryCircle]

CircleEquation[ParryCircle]

Dimensions[{{1,2,3}}]

Derivation

Factor[Together//@CircleEquation[{X[2],X[23],X[111]}]]

eqn=Numerator[%]

FullSimplify[Together//@CircleFunction[eqn]]

CircleEquation[{CyclicTrilinears[-((b c (-2 (a^2)+(b^2)+(c^2)))/(3 ((a^2)-(b^2)) ((a^2)-(c^2))))]}]

Center

CircleCenter[ParryCircle]

FullSimplify[(Together//@DistanceTrilinears[CircleCenter[ParryCircle]])((6a b c(a^2-b^2)(b^2-c^2)(c^2-a^2))/((-a+b+c) (a-b+c) (a+b-c) (a+b+c)))]

IncidentCenters[KimberlingCenters,CircleCenter[ParryCircle]]//Timing

CenterName[X[351]]

CenterFunction[X[351]]

Center Function

CircleFunction[ParryCircle]

InputForm[%]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[ParryCircle]]]//Timing

CenterFunction[X[690]]

CenterName[X[690]]

Radius

rr=FullSimplify[DistanceTrilinears/@CircleRadius[ParryCircle],TriangleQ[]]

FullSimplify[Abs[(a^2-b^2)(a^2-c^2)(b^2-c^2)],TriangleQ[]]

Factor[a b c ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2)))]

ToAngles[%]

CircleRadius[ParryCircle]

SA:=(b^2+c^2-a^2)/2;
SB:=(a^2+c^2-b^2)/2;
SC:=(a^2+b^2-c^2)/2;

((a b c)/3)Abs[Factor[(1/(SA-SB))+(1/(SB-SC))+(1/(SC-SA))]]

Simplify[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)>=0,TriangleQ[]]

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,CircleEquation[ParryCircle]])//Timing

{#,CenterName[#]}&/@ctrs//TableForm

Orthogonality

Check

Together//@DistanceTrilinears[First[OrthogonalQ[BrocardCircle,ParryCircle]]]//Timing

Graphic

Through[circs[t]]

t=Triangle[{{0,0},{1,0},{-.4,.7}}];
circs={BrocardCircle,Circumcircle,ParryCircle,LucasCirclesRadicalCircle,LucasInnerCircle};
centers={"(Subscript[X, 182])","O","(Subscript[X, 351])","(Subscript[X, 1151])","(Subscript[X, LIS])"};
colors={Red,Orange,Green,Blue,Purple};
Show[Graphics[{
t,
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{Center[#[t]]&/@circs,centers,colors}],
Transpose[{VertexPoints[t],{"A","B","C"}}]
],LabelOffset->.06
]
}],AspectRatio->Automatic,PlotRange->{{-1.2,1.2},{-.6,1.5}}]